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AUTHOR Kieffer, Kevin M.TITLE Orthogonal versus Oblique Factor Rotation: A Review of the

Literature regarding the Pros and Cons.PUB DATE 1998-11-04NOTE 32p.; Paper presented at the Annual Meeting of the Mid-South

Educational Research Association (27th, New Orleans, LA,November 4-6, 1998).

PUB TYPE Book/Product Reviews (072) Speeches/Meeting Papers (150)EDRS PRICE MF01/PCO2 Plus Postage.DESCRIPTORS *Factor Analysis; Heuristics; Literature Reviews; Oblique

Rotation; Orthogonal Rotation; *Scores; Tables (Data);*Validity

IDENTIFIERS Exploratory Factor Analysis; *Orthogonal Comparison;*Rotations (Factor Analysis)

ABSTRACTFactor analysis has been characterized as being at the heart

of the score validation process. In virtually all applications of exploratoryfactor analysis, factors are rotated to better meet L. Thurstone's simplestructure criteria. Two major rotation strategies are available: orthogonaland oblique. This paper reviews the numerous rotation options available inthe factor analysis literature, examining the pros and cons of variousanalytic choices. A heuristic data set was examined to make the discussionconcrete. Some guidelines are also offered for resolving differences in theanalytic choices so that the appropriate rotation methods can be selected.(Contains 10 tables and 16 references.) (Author/SLD)

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Factor Rotation 1

Running head: ROTATION STRATEGIES

Orthogonal Versus Oblique Factor Rotation: A Review of the

Literature Regarding the Pros and Cons

Kevin M. Kieffer

Texas A&M University 77843-4225

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Points of view or opinions stated in thisdocument do not necessarily representofficial OERI position or policy.

Paper presented at the annual meeting of the Mid-SouthEducational Research Association, New Orleans, November 4, 1998.

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Factor Rotation 2

Abstract

Factor analysis has been characterized as being at the heart of

the score validation process. In virtually all applications of

exploratory factor analysis, factors are rotated to better meet

Thurstone's simple structure criteria. Two major rotation

strategies are available: orthogonal and oblique. The present

paper reviewed the numerous rotation options available in factor

analysis, and in particular examined the pros and cons of various

analytic choices. A heuristic data set was examined to make the

discussion concrete. Further, some guidelines are offered for

resolving differences in the analytic choices so that the

appropriate rotation methods can be selected.

Factor Rotation 3

Orthogonal Versus Oblique Factor Rotation: A Review of the

Literature Regarding the Pros and Cons

The utilization of factor analytic techniques in the social

sciences has been integral to the development of theories and the

evaluation of the construct validity of scores. It is not

surprising, therefore, that Pedhazur and Schmelkin (1991, p. 66)

noted, "Of the various approaches to studying the internal

structure of a set or indicators, probably the most useful is some

variant of factor analysis." The current accessibility and

widespread applicability of factor analytic techniques have

rendered these methods popular among researchers exploring data

structure and the construct validity of scores.

One variant of factor analysis is termed exploratory factor

analysis. In this application of factor analysis, the primary

concern is with the development of theories or the generation of

alternative explanations for commonly accepted theories about a

phenomenon of interest. As noted by Tinsley and Tinsley (1987),

Factor analysis is an analytic technique that permits

the reduction of a large number of interrelated

variables to a smaller number of latent or hidden

dimensions. The goal of factor analysis is to achieve

parsimony by using the smallest number of explanatory

concepts to explain the maximum amount of common

variance in a correlation matrix. (p. 414)

Thus, EFA is a data reduction technique that permits the reduction

of a large number of variables (e.g., test items, individuals)

into constituent components by examining the amount of variance

that can be reproduced by the latent or synthetic variables

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Factor Rotation 4

underlying the observed or measured variables.

The purpose of the present paper was to briefly explicate the

concept of exploratory factor analysis with an emphasis on a

conceptual understanding of the factor rotation process and the

two types of rotation strategies available to researchers.

Examples were provided using the Holzinger and Swineford (1939)

data set that has been utilized extensively in illustrating factor

analytic principles (cf. Gorsuch, 1983). Further, the relative

merits of utilizing different rotation strategies were explored,

and guidelines for their appropriate application were provided.

Brief Overview of Exploratory Factor Analysis

Factor analysis has been conceptually available to

researchers since the turn of the century (Thompson & Daniel,

1996), but unfortunately due to the complex nature of the

mathematical manipulations necessary to perform computations, has

not been used extensively until the advent of the computer.

However, a renewed interest in factor analytic techniques was

evidenced in the middle of the 20th century following the

delineation of test standards by the American Psychological

Association (Thompson & Daniel, 1996). Following the introduction

of these test standards, researchers began utilizing factor

analysis to demonstrate the validity of scores generated by their

instruments.

EFA has generally been utilized for two general purposes in

the social sciences. The first general purpose has been to better

understand the structure of a data set when no previous

information on the data structure is available. This is an example

of utilizing EFA as a tool in the generation of theories about

Factor Rotation 5

phenomena of interest, and is an example of an appropriate

application of this analytic technique. The second general

application of EFA has been to reexamine patterns in data sets

when the tenability of the emergent factors in previous research

has been questioned. The latter example use of EFA is

inappropriate, as a model-to-data fit can be directly evaluated

through another variant of factor analysis termed confirmatory

factor analysis (CFA). Thus, the primary application of EFA is to

explore the factor structure of a set of indicators (e.g.,

variables, test items, individuals, occasions) when no previous

research is available.

Basics of Exploratory Factor Analysis

Factor rotation is just one of several steps completed when

conducting an EFA. Further, each individual step often involves a

decision based on the intrinsic values of the researcher or the

goals of the analysis. Consequently, there are many different ways

in which to conduct a factor analysis, and each different approach

may render distinct results when certain conditions are satisfied

(cf. Gorsuch, 1983). The one consistent element in conducting an

EFA, however, is that the results of the analysis are based solely

on the mechanics and mathematics of the method and not on the a

priori theoretical considerations of the researcher (Daniel,

1989). To provide a context for the discussion of factor rotation,

a few of the steps involved in conducting a factor analysis are

briefly reviewed.

Matrices of Association. One of the first decisions to be

made in performing an EFA is to determine the manner in which the

data matrix will be represented in the analysis. Since all

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Factor Rotation 6

statistical analyses are correlational (Cohen, 1968; Knapp, 1978;

Thompson, 1997a), the focus of every statistical analysis is on

the relationship among a set of variables or other entities (e.g.,

people) that may be factored. Matrices of association (e.g.,

correlation matrices, variance-covariance matrices) are arrays of

numbers that are utilized to concisely express the linear

relationships between a larger set of variables. The most common

matrix of association utilized in EFA is the correlation matrix

(in which values of 1.0 are on the main diagonal and bivariate

correlation coefficients between the variables are on the off-

diagonals), perhaps partly because this is the default in most

statistical software packages.

Factor Extraction. After the researcher has chosen which

matrix of association will be utilized in the analysis, the

researcher must then determine which extraction method to employ

in conducting the analysis. Factor extraction refers to removing

the common variance that is shared among a set of variables. There

are currently several different techniques available for the

extraction of common variance (e.g., principal components analysis

and principal factors analysis), and the results generated by the

analysis can differ based on the particular method of extraction

utilized.

Of the techniques available, principal components analysis

and principal factors analysis are the two most widely used

extraction methods in EFA. Although some researchers have argued

that the difference between these extraction methods is negligible

(cf. Thompson, 1992), other researchers have contended that the

difference is substantial enough to warrant careful consideration

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Factor Rotation 7

of the extraction method utilized (cf. Gorsuch, 1983). The

interested reader is referred to Gorsuch (1983), Stevens (1996) or

Tinsley and Tinsley (1987) for a more thorough treatment of this

material.

Factor and Coefficient Generation. One advantage in employing

a factor analysis is that each latent or synthetic variable

(factor) extracted from the analysis is perfectly uncorrelated

with all of the other factors. This is often advantageous when the

purpose of the EFA is theory generation as the interpretation of

the extracted factors is thereby greatly simplified.

When extracting factors, only a certain portion of the

variance for any given variable will be reproduced by the factors.

Two matrices are formed as a result of the factor extraction

procedure, the factor pattern matrix (comprised of weights that

are identical to B weights in multiple regression analysis and

that indicate the relative importance of a given variable to the

extracted factors with the influence of the other variables

removed) and the factor structure matrix (coefficients that

represent the bivariate correlation of measured/observed variables

with scores on the extracted latent/synthetic factors). Since the

extracted factors are always initially perfectly uncorrelated, the

factor pattern matrix and the factor structure matrix are exactly

equal; thus these two matrices can be simplified into one matrix

termed the "factor pattern/structure matrix."

Variance-Accounted-For Statistics

After the factor pattern/structure matrix is generated, it is

possible to derive two variance-accounted-for statistics that help

the researcher determine the amount of variance that is reproduced

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Factor Rotation 8

by the latent constructs. The first of these is the communality

coefficient, h2, which can be defined as the amount of variable

variance that is reproduced by the factors. This value is

calculated by summing the squared pattern/structure coefficients

across the row for each variable. The resultant coefficient is an

index of the proportion of total variance for a given variable

that is reproduced by the extracted factors. Since it is a squared

statistic, it can range from 0 to 1.0.

Another variance-accounted-for statistic that is generated

from the factor pattern/structure matrix is the eigenvalue. An

eigenvalue represents the amount of variance in the original data

set that is reproduced by a given factor. For the principal

components case, eigenvalues can be computed by summing the

squared factor pattern/structure coefficients down the columns of

the matrix. Eigenvalues represent the amount of factor-reproduced

variance and their values can range from 0.0 to the total number

of variables in the analysis. Eigenvalues can also serve as effect

size measures, as each eigenvalue can be divided by the number of

total variables in the analysis and a percentage of the total

variance for a given factor can be computed.

Factor Retention. After the factor pattern/structure matrices

and variance-accounted-for statistics and have been computed, the

researcher must decide the number of factors to

analysis.

divergent

Since different retention methods can

results, it is generally important to

retain in the

often generate

examine more than

one factor retention method.

Two popular methods of determining the number of factors to

retain is the eigenvalue greater than 1.0 rule (Kaiser, 1960) and

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Factor Rotation 9

the scree test (Cattell, 1966). It is important to carefully

examine all of the eigenvalues, however, as previous research has

reported that in certain situations the eigenvalues greater than

1.0 rule and scree test can underestimate or overestimate the

number of factors that should be retained (cf. Hetzel, 1996; Zwick

& Velicer, 1986).

Interpretation of Results. After the appropriate number of

factors are retained in the analysis, it is necessary to interpret

the results. It is often difficult to interpret the initial

factor/pattern structure matrix as many of the variables typically

manifest noteworthy coefficient magnitudes on many of the retained

factors (coefficients greater than 10.601 are often considered

large and coefficients of 10.351 are often considered moderate)

and especially on the first factor.

Rotation of Factor Analytic Results

After the factor analysis is completed, it is usually

necessary to rotate the factors to formulate a better solution

that is more interpretable (i.e., has better "simple structure"

(Thurstone, 1947)). Thompson (1984, pp. 31-34) demonstrated how

the unrotated pattern/structure matrix actually misrepresents the

true nature of the factors, and how factor rotation resolves this

misrepresentation. Pedhazur and Schmelkin (1991) commented on the

need to rotate factors:

While the results of a factor analysis may produce a

good fitting solution it is not necessarily susceptible

to a meaningful interpretation. It is in attempts to

improve the interpretability of results that factors are

rotated.... Because there is an infinite number of ways

1 0

Factor Rotation 10

in which factors can be transformed or rotated, the

question arises: Are some rotations better than

others?.... The reason that rotations are resorted to

for the purpose of improving interpretability of factor

analytic results and interpretability, is by its very

nature, inextricably intertwined with theory. (p. 611)

Interpretation of the factor analytic results is therefore

almost always aided by the rotation of the factor solution, as it

is possible to redistribute the common variance across the factors

to achieve a more parsimonious solution. It is important to note,

however, that factor rotation is not "cheating" and does not

generate or discover more common variance; rather, factor rotation

merely redistributes the variance that has been previously

explained by the extracted factors.

After a factor solution is rotated, the first unrotated

factor may not account for the largest portion of the variance and

thus may not have the largest variance-accounted-for value. Since

the variance has been redistributed throughout the factors, any of

the factors could account for the largest proportion of the total

variance. Additionally, after the rotation is conducted,

eigenvalues are no longer termed as such; rather, after rotation,

the variance-accounted-for statistic for the factors (columns of

the factor pattern/structure matrix) is termed "trace." One of the

most commonly committed mistakes by researchers is believing that

the eigenvalue for a given factor before extraction informs

judgement regarding the variance accounted for by the factor after

the factor solution is rotated (Hetzel, 1996).

Objectives of Factor Rotation. The objective of factor

1

Factor Rotation 11

rotation is to achieve the most parsimonious and simple structure

possible through the manipulation of the factor pattern matrix.

Thurstone's (1947) guidelines for rotating to simple structure

have largely influenced the development of various rotational

strategies. The most parsimonious solution, or simple structure,

has been explained by Gorsuch (1983, pp. 178-179) in terms of five

principles of factor rotation:

1. Each variable should have at least one zero loading.

2. Each factor should have a set of linearly independent

variables whose factor loadings are zero.

3. For every pair of factors, there should be several

variables whose loadings are zero for one factor but

not the other.

4. For every pair of factors, a large proportion of

variables should have zero loadings on both factors

whenever more than about four factors are extracted.

5. For every pair of factors, there should only be a

small number of variables with nonzero loadings on both.

Thus, factor rotation is devised to shift the factors in their

factor space so that each variable in the analysis has a large

factor pattern coefficient on only one factor and has very small

or zero factor pattern coefficients on the other extracted latent

constructs.

The primary goal in rotating to simple structure is to

produce better fitting solutions that are more replicable across

studies. As stated by Gorsuch (1983), "Thurstone showed that

[simple structure] rotation leads to a position being identified

for each factor that would be independent of the number of

1 2

Factor Rotation 12

variables defining it. Therefore, a simple structure factor should

be relatively invariant across studies" (p. 177).

Types of Factor Rotation

Orthogonal Factor Rotation

Two types of factor rotation are available: orthogonal and

oblique. Orthogonal rotation shifts the factors in the factor

space maintaining 90 degree angles of the factors to one another

to achieve the best simple structure. Since the cosine of the

angles between vectors of unit length equals r, and the cosine of

a 90 degree angle is zero, this rotation strategy maintains the

perfectly uncorrelated nature of the factors after the solution is

rotated and often aids in the interpretation process since

uncorrelated factors are easier to interpret. In theory, the

results of an orthogonal rotation are likely to be replicated in

future studies since there is less sampling error in the

orthogonal rotation due to less capitalization on chance that

would occur if more parameters were estimated, as is the case in

oblique rotation.

Varimax Rotation. Of the orthogonal rotation strategies

available, one of the most popular orthogonal rotation technique

is rotation to the varimax criterion developed by Kaiser (1960).

In this technique the factors are "cleaned up" so that every

observed variable has a large factor pattern/structure coefficient

on only one of the factors. Varimax rotation produces factors that

have large pattern/structure coefficients for a small number of

variables and near-zero or very low pattern/structure coefficients

with the other group of variables.

Quartimax Rotation. Another popular orthogonal rotation

13

Factor Rotation 13

strategy is quartimax. In this rotation strategy, the factor

pattern of a variable is simplified by forcing the variable to

correlate highly with one main factor (the so-called G factor of

early IQ studies) and very little with the remaining factors

(Stevens, 1996). The variables are much easier to interpret in

this case, but the factors are more difficult to interpret since

all variables are primarily associated with a singe factor.

Since the emphasis of varimax rotation is on easing the

interpretability of the factors, it may seem logical to utilize

varimax rotation when employing an orthogonal rotation strategy.

However, researchers must carefully examine the objectives of an

analysis, as quartimax rotation would be preferred in an analysis

in which the researcher expected one general (i.e., the so-called

"G") factor. Although the results between quartimax and varimax

tend to be similar (Stevens, 1996), thoughtful researchers will

examine their expectations of the analysis and choose orthogonal

rotation strategies accordingly.

Advantages and Limitations of Orthogonal Rotation. There are

several advantages to employing orthogonal rotation strategies.

First, the factors remain perfectly uncorrelated with one another

and are inherently easier to interpret. Secondly, the factor

pattern matrix and the factor structure matrix are equivalent and

thus, only one matrix of association must be estimated. This means

that the solution is more parsimonious (i.e., fewer parameters are

estimated) and thus, in theory, is more replicable.

Orthogonal rotation strategies do, however, have limitations.

Orthogonal rotations often do not honor a given researcher's view

of reality as the researcher may believe that two or more of the

14

Factor Rotation 14

extracted and retained factors are correlated. Secondly,

orthogonal rotation of factor solutions may oversimplify the

relationships between the variables and the factors and may not

always accurately represent these relationships.

Oblique Factor Rotation

The second type of factor rotation is termed oblique

rotation. This method of rotation provides for correlations among

the latent constructs. This rotation strategy is termed oblique

because the angles between the factors becomes greater or less

than the 90 degree angle.

Direct Oblimin. One type of oblique rotation strategy is

direct oblimin. This strategy is moderated by a delta value

(chosen by the researcher), in which positive values of delta

produce higher correlations between factors and negative values of

delta produce smaller correlations between factors. The direct

oblimin strategy more closely honors the nature of reality and

demands careful consideration by the researcher, as the

correlation between the factors must be set prior to analysis.

The preset default value of delta can have serious

consequences on the results of the direct oblimin factor rotation.

Gorsuch (1983) indicated that at delta = -4 the factors actually

become orthogonal, and he further recommended varying delta from 1

to -4 to help researchers ascertain a clearer conception of their

data structure. Researchers have objected to the use of this

strategy, however, as the technique appears very subjective in

nature (regarding the preset correlation of the factors).

Promax. One of the most popular oblique rotation strategies

is promax (see Hetzel, 1996). In this technique, the researcher is

Factor Rotation 15

attempting to achieve the most parsimonious simple structure given

that the factors are allowed to be correlated with one another.

Promax rotation has three distinct steps, the first of which is to

rotate the factors orthogonally. Next, a target matrix is

contrived by raising the factor pattern/structure coefficients to

an exponent greater than two (typically exponents of three or four

are used). The coefficients in the target matrix become smaller,

but the absolute distance between them actually increases. For

example, if 0.9 and 0.3 are two factor pattern/structure

coefficients, the first coefficient is three times larger than the

second; however, if both values are squared, the resultant value

of 0.81 is nine times larger than 0.09. Thus, the result of

transforming the factor pattern/structure matrix is that many

moderate coefficients (i.e., 10.301 or smaller) approach zero more

quickly than the large coefficients (i.e., 10.601 or larger).

The final step in a promax rotation involves the

"Procrustean" rotation of the original matrix to a best fit

position with the target matrix. Promax is often the oblique

rotation strategy of choice, as it is relatively easy to use,

typically provides good solutions, and tends to generate more

replicable results than the direct oblimin rotations.

Advantages and Limitations of Oblique Rotations. Oblique

rotation strategies can be useful to researchers for a variety of

reasons. One advantage of using an oblique rotation strategy is

that the solution more closely honors the researcher's view of

reality. Unfortunately, oblique rotations may be difficult to

interpret, especially if there is a high degree of correlation

among the factors. Since the factor pattern and factor structure

16

Factor Rotation 16

matrices are not equal, both have to be interpreted in conjunction

with the other.

However, an oblique factor solution inherently tends to be

less parsimonious. For example, if 5 factors for 100 factored

entities (e.g., variables) are extracted and orthogonally rotated,

only 500 factor pattern/structure coefficients are estimated (the

5 x 5 factor correlation matrix is not estimated, since it is

constrained to have l's on the diagonal and O's everywhere else).

If the same EFA factors are rotated obliquely, 1,010 coefficients

(500 factor pattern coefficients, plus 500 factor structure

coefficients, plus 10 factor correlation coefficients (the 10 non-

redundant off-diagonal entries in the 5 x 5 factor correlation

matrix)) are estimated. [It might be argued, however, that only

510 coefficients are estimated in this case, since with either the

10 unique factor correlation coefficients, and either the 500

pattern or the 500 structure coefficients, the remaining 500

pattern or structure coefficients are fully determined.]

The fact that more parameters are estimated in an oblique

rotation means that oblique solutions almost always better fit

sample data than do orthogonal solutions. However, some of this

fit involves "overfitting" sampling error variance. This means

that orthogonal solutions, though they may tend to somewhat fit

sample data less well, are generally more replicable in future

samples, since orthogonal solutions capitalize on less sampling

error. Usually, at least in EFA, somewhat poorer fit is deemed an

acceptable tradeoff for better solution replicability (i.e.,

factor invariance). As the degree of correlation between the

factors decreases, both orthogonal and oblique solutions will tend

17

Factor Rotation 17

to provide increasingly similar results. Given that oblique

solutions are less parsimonious and therefore less replicable, an

oblique rotation would therefore only be employed when the

benefits of simpler, more interpretable structure outweigh the

costs of less replicability (i.e., when the orthogonal factors are

not readily interpretable, and the oblique factors are fairly

highly correlated but more interpretable).

Which Type of Rotation is Better?

The decision to rotate orthogonally or obliquely is often

difficult for researchers and is largely based on the goal of the

analysis. If the goal of the analysis is to generate results that

best fit the data, then oblique rotation seems to be the logical

choice. Conversely, if the replicability of the factor analytic

results is the primary focus of the analysis, then an orthogonal

rotation might be preferable since results from orthogonal

rotation tend to be more parsimonious. The decision to rotate

orthogonally or obliquely was discussed by Pedhazur and Schmelkin

(1991):

the decision whether to rotate factors orthogonally or

obliquely reflect's one's conception regarding the

structure of the construct under investigation. It boils

down to the question: Are aspects of the a postulated

multidimensional construct intercorrelated?... The

preferred course of action is, in our opinion, to rotate

both orthogonally and obliquely. When, on the basis of

the latter, it is concluded that the correlations among

the factors are negligible, the interpretation of the

simpler orthogonal solution becomes tenable. (p. 615)

18

Factor Rotation 18

Researchers (Thurstone, 1947; Cattell, 1966; 1978) have

challenged the utility of orthogonal rotation in preference for

the utilization of oblique strategies. Thurstone (1947, P. 139)

contended that the use of orthogonal rotation indicates "...our

ignorance of the nature of the underlying structure... The reason

for using uncorrelated [factors] can be understood, but it cannot

be justified." Similarly, Cattell (1978, p. 128) argued in regard

to researchers performing orthogonal rotation, "...in half of

[the] cases it is done in ignorance of the issue rather than by

deliberate intent." Consequently, even though orthogonal rotation

eases the interpretability of the factor solution, it may not

accurately portray the relationships between the variables and the

emergent factors. However, many researcher do not blindly employ

orthogonal strategies and instead utilize these approaches to ease

result interpretation and to generate a more parsimonious result.

Two elements typically whether orthogonal and oblique

rotation strategies will generate similar or identical results:

(a) the factor to variable ratio; and (b) the degree of

correlation between the factors. When the ratio of variables to

factors is small, both rotation strategies will produce similar

results, as simple structure will tend to be the same regardless

of the type of rotation. Further, if the correlation between the

factors is small (i.e., factor correlation coefficients closer to

zero), then orthogonal and oblique rotation strategies will

generally produce similar, if not identical, results.

In sum, choosing a rotation strategy to employ in factor

analysis is not an arbitrary decision; rather, the appropriate

choice of either an orthogonal or oblique rotation largely depends

19

Factor Rotation 19

on the goals of the analysis (best fit to data or replicability of

the analysis), the factor to variable ratio, and the degree of

correlation between the factors. The guidelines stated by Pedhazur

and Schmelkin (1991) appear to contend appropriately with the

issues, as these authors suggested conducting both strategies and

then comparing the results of the different rotations. If the

difference between the two results is negligible, then the

researcher can interpret the orthogonal rotation. Conversely, if

the differences between the two rotations is noteworthy, then the

researcher must consider interpreting the oblique rotation.

Heuristic Example of Differences in Rotation Strategies

To demonstrate the differences in orthogonal and oblique

rotation strategies, a heuristic example was utilized in which

tests Tl, T8, T9, Tll, T12, T14, T15 and T16 from the original

Holzinger and Swineford (1939) data set were factor analyzed.

Further, the factor pattern and structure matrices in each of

three cases (no rotation, orthogonal rotation and oblique

rotation) were compared. The means, standard deviations and labels

for the raw data are presented in Table 1 and the variable

correlation matrix is presented in Table 2.

Insert Table 1 and 2 About Here.

Principal components analysis (PCA) was chosen to extract the

variables since this method yields results similar to principal

factors analysis as the number of factored entities increases

(Thompson & Daniel, 1996). The results of the factor extraction

are presented in Tables 3 and 4.

9 0

Factor Rotation 20


Notice that two matrices are contrived as a result of the

PCA: one is a factor pattern matrix (Table 3) and the other is a

factor structure matrix (Table 4). Since PCA was utilized to

extract the factors, the extracted factors are perfectly

uncorrelated and, consequently, the factor pattern and structure

matrices are exactly equal. Thus, the factor pattern matrix and

the factor structure matrix can be combined into one factor

pattern/structure matrix since all of the values are identical and

no information will be lost.

The factor correlation matrix is presented in Table 5. The

table indicates that each of the factors correlates perfectly with

itself but does not correlate with any of the other factors (i.e.,

the factors are perfectly uncorrelated). This is the universal

result of initial factor extraction.

Insert Table 5 About Here.

An examination of Table 4 reveals that the factor saturation

(which observed variables have large coefficients on which latent

constructs) is complex, and it is difficult to interpret the

factor pattern/structure matrix in its present form. In Table 4,

all eight of the variables have pattern/structure coefficients

greater than 1.351, the criterion generally indicative of a

moderately large coefficient. Thus, to more easily interpret the

results, the three factor solution could be rotated to different

criteria. The three-factor solution was first rotated to the

varimax criterion. The results of the varimax rotated solution are

21

Factor Rotation 21

presented in Table 6.


An examination of the results presented for the four factor

solution rotated to the varimax criterion reveals that

ascertaining which variables are associated with which factors has

been greatly facilitated by the rotation procedure. Factor I is

most highly saturated with tests Tl, T8, and T9 (perhaps

indicating a strong verbal component to the tests). Factor II is

most highly saturated with tests T14, T15 and T16 (likely

indicating a strong memory component to those three tests).

Finally, Factor III is most highly saturated with T11 and T12

(appearing to indicate a speed component common to both tests).

Thus, the rotation of the results to the varimax criterion enabled

the easier interpretation of the results.

It is important to notice that the communality coefficients

for the varimax rotated solution are identical to the communality

coefficients in the unrotated four factor solution. The reason is

that the variable variance reproduced by a given factor is only

redistributed in the rotated solution, and no new variance is ever

generated through a rotation procedure.

The first factor still accounts for the majority of the

variance after rotation, even though after rotation other factors

could account for the majority of the variance. It is important to

note that the total variance-accounted-for the by the three-factor

solution before rotation (64.4%) is exactly equal to the total

variance-accounted-for after rotation. Notice, however, that the

factors are still perfectly uncorrelated as presented in Table 7.

2 ?

Factor Rotation 22


The unrotated factor pattern and factor structure matrices in

Tables 3 and 4 were rotated to the direct oblimin criterion with

delta equal to zero (see Gorsuch, 1983 for a more detailed

explanation of the effects of varying the value of delta) to

compare the parsimony of the two rotation strategies. The results

of the oblique rotation are presented in Tables 8 and 9.


When interpreting the results of an oblique rotation, it is

necessary to interpret two separate factor association matrices

since the factor pattern matrix is no longer identical to the

factor structure matrix. Similarly to multiple linear regression,

however, it is critically important to interpret both pattern

coefficients (standardized weights) and structure coefficients as

each can provide only one piece of information regarding the

larger relationship (Thompson, 1997b).

By examining the variance-accounted-for by each factor

(trace), the results of the direct oblimin rotation appear to

closely resemble the results generated by the varimax rotation. As

a matter of fact, the oblique rotation strategy only increased the

variance-accounted-for by the first factor by one-tenth of a

percent. Thus, the results of orthogonal and oblique rotation

strategies in this case are essentially equal and will lead to

almost identical conclusions. However, there are some issues to

consider in interpreting the results of the oblique rotation.

2 3

Factor Rotation 23

Since the factors are allowed to be correlated in the oblique

case, it is important to consider the correlation of the factors

when interpreting the solution. The factor correlation matrix for

the direct oblimin rotation is presented in Table 10. Factors I

and III share approximately 10% common variance and factors I and

II share over 5% of their variance. Since the common variance

between the factors is minimal in this case, the results of the

oblique rotation would be interpreted much in the same manner as

the orthogonal rotation. Since both solutions are similar and the

orthogonal rotation is interpretable and more parsimonious, most

researcher would select this as the preferred solution.


Other items in the oblique rotation deserve comment as well.

Notice that the communality coefficients in the oblique case are

exactly equal to the results attained in the unrotated and varimax

rotated factor solutions, thus again illustrating only the

redistribution of common variance. The sum of the communality

coefficients is still equal to 64.4%, as in all of the prior

analyses. The trace and communality coefficients are computed

slightly differently, however, as it is necessary to multiple a

given factor pattern coefficient by the corresponding factor

structure coefficient and then to sum down the columns or across

the rows to derive the various variance-accounted-for estimate

(see the note on Tables 8 and 9 for a more detailed explanation).

Summary of Heuristic Example

The primary difference between these two rotation strategies

in this heuristic example is the correlation between the factors,

2 4

Factor Rotation 24

and one of the difficulties incurred in interpreting oblique

rotations is how to explicate a high degree of correlation among

the factors. Most of the correlations in the present example were

relatively small and posed little difficulty in the present

analysis. However, in other analyses a high degree of correlation

between the factors could result in very different interpretations

of the factor solutions between the orthogonal and oblique cases.

Summary of Exploratory Factor Analysis and Rotation Strategies

The present heuristic example of EFA has demonstrated the

differences (or lack thereof) typically encountered in EFA. Using

orthogonal versus oblique rotation strategies demands careful

consideration, and each strategy may be preferable in certain

situations. However, as the factor to variable ratio and the

factor correlation decrease, the results from orthogonal and

oblique rotations tend to become more similar. Thus, the

researcher must decide, based on intrinsic judgments and

expectancies of the analysis, which rotation strategy will

generate the most appropriate results.

23

Factor Rotation 25

References

Cattell, R.B. (1966). The scree test for the number of

factors. Multivariate Behavioral Research, 1, 245-276.

Cattell, R.B. (1978). The scientific use of factor analysis

in behavioral and life sciences. New York: Plenum.

Cohen, J. (1968). Multiple regression as a general

data-analytic system. Psychological Bulletin, 70, 426-433.

Gorsuch, R.L. (1983). Factor analysis (2nd ed.). Hillsdale,

NJ: Erlbaum.

Hetzel, R.D. (1996). A primer on factor analysis with

comments on patterns of practice and reporting. In B. Thompson

(Ed.), Advances in social science methodology (Vol. 4, pp. 175-

206). Greenwich, CT: JAI Press.

Holzinger, K.J., & Swineford, F. (1939). A study in factor

analysis: The stability of a bi-factor solution (No. 48). Chicago:

University of Chicago.

Kaiser, H.F. (1960). The application of electronic computers

to factor analysis. Educational and Psychological Measurement, 20,

141-151.

Knapp, T.R. (1978). Canonical correlation analysis: A general

parametric significance testing system. Psychological Bulletin,

85, 410-416.

Pedhazur, E.J., & Schmelkin, L.P. (1991). Measurement,

design, and analysis: An integrated approach. Hillsdale, NJ:

Erlbaum.

Stevens, J. (1996). Applied multivariate statistics for the

behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

Thompson, B. (1984). Canonical correlation analysis: Uses and

2R

Factor Rotation 26

interpretation. Thousand Oaks, CA: Sage.

Thompson, B. (1997a, August). If statistical significance

tests are broken/misused, what practices should supplement or

replace them? Invited paper presented at the annual meeting of the

American Psychological Association, Chicago.

Thompson, B. (1997b). The importance of structure

coefficients in structural equation modeling confirmatory factor

analysis. Educational and Psychological Measurement, 57, 5-19.

Thompson, B., & Daniel, L.G. (1996). Factor analytic evidence

for the construct validity of scores: A historical overview and

some guidelines. Educational and Psychological Measurement, 56,

197-208.

Thurstone, L.L. (1947). Multiple factor analysis. Chicago:

University of Chicago Press.

Tinsley, H.E.A., & Tinsley, D.J. (1987). Uses of factor

analysis in counseling psychology research. Journal of Counseling

Psychology, 34, 414-424.

Zwick, W.R., & Velicer, W.F. (1986). Comparison of five rules

for determining the number of components to retain. Psychological

Bulletin, 99, 432-442.

Table 1

Means and

Swineford,

Standard Deviation of 8

(1939)

Factor Rotation 27

Variables From Holzinger and

Variable Mean SD LabelTl 29.61462 7.00459 Visual Perception TestT8 26.12625 5.67544 Word ClassificationT9 15.29900 7.66922 Worded Meaning TestTll 69.16279 15.67025 Speeded Code TestT12 110.54153 20.25230 Speeded Counting DotsT14 175.15282 11.50753 Memory of Target WordsT15 90.00997 7.72937 Memory of Target NumbersT16 102.52492 7.63306 Memory of Target Shapes

Table 2

Correlation Matrix for Example Data

Tl T8 T9 Tll T12 T14 T15 T16

TlT8T9TllT12T14T15T16

1.0000.3310.3568.2859.2239.1289.1845.3646

1.0000.5816.3133.1842.1786.0527.2924

1.0000.2902.1496.1721.0519.2528

1.0000.3977.2248.1400.3048

1.0000.0385.0779.1463

1.0000.3967.3875

1.0000.3382 1.0000

Table 3

Unrotated Factor Pattern Matrix

Factor Rotation

Variable I II III h2

Tl .63601 -.13656 -.02894 .42399T8 .66835 -.38257 -.35789 .72114

T9 .64966 -.39131 -.41078 .74392

Tll .64121 -.15033 .42783 .61679T12 .43541 -.27421 .73125 .79949

T14 .51078 .58907 -.12257 .62293

T15 .41950 .68773 .07930 .65524

T16 .67430 .33096 -.05772 .56755

(Sum =)

Eigenvalues 2.76551 1.34543 1.04007 5.15115

% of Variance 34.6 16.8 13.0 64.4

Table 4

Unrotated Factor Structure Matrix


T1 .63601 -.13656 -.02894 .42399

T8 .66835 -.38257 -.35789 .72114

T9 .64966 -.39131 -.41078 .74392

T11 .64121 -.15033 .42783 .61679

T12 .43541 -.27421 .73125 .79949

T14 .51078 .58907 -.12257 .62293

T15 .41950 .68773 .07930 .65524

T16 .67430 .33096 -.05772 .56755

(Sum =)

Eigenvalues 2.76551 1.34543 1.04007 5.15115

% of Variance 34.6 16.8 13.0 64.4

28

Table 5

Factor Correlation Matrix

Factor 1 1.00000Factor 2 .00000 1.00000Factor 3 .00000 .00000 1.00000

29

Factor Rotation

Table 6

Factor Pattern/Structure Matrix Rotated to the Varimax Criterion

Variable 1 II III h2

Tl .52093 .23462 .31236 .42399T8 .83904 .06217 .11527 .72114T9 .85882 .04716 .06422 .74392Tll .28515 .20428 .70267 .61679T12 .03921 -.02612 .89291 .79949T14 .13507 .77746 -.01540 .62293T15 -.08429 .80052 .08542 .65524T16 .33755 .64787 .18403 .56755

(Sum =)Trace 1.93514 1.76860 1.44741 5.15115% of Variance 24.2 22.1 18.1 64.4

29

Note. Coefficients greater than 1.351 are underlined.

Table 7

Factor Correlation Matrix After Rotation to Varimax Criterion

Factor 1 1.00000Factor 2 .00000 1.00000Factor 3 .00000 .00000 1.00000

Factor Rotation 30

Table

Factor

8

Pattern Matrix Rotated to the Oblimin Criterion

Variable h2

Tl .46919 .16137 .23464 .42399

T8 .85781 -.03373 -.00270 .72114

T9 .88892 -.04698 -.05788 .74392

Tll .15958 .12135 .68095 .61679

T12 -.10360 -.10761 .93555 .79949

T14 .05184 .79198 -.10216 .62293

T15 -.19885 .83029 .03285 .65524

T16 .24771 .61946 .09006 .56755

Note. Communality coefficients are now computed differently. Tocompute the h2 for variable Tl, each pattern coefficient ismultiplied by its corresponding structure coefficient and thensummed across the rows. For Tl, h2 = (.46919)(.58274) +(.16137)(.32857) + (.23464)(.41574) = .42399. The new communalitycoefficient is identical to the value attained in the Table 4

analysis.

Table 9

Factor Structure Matrix Rotated to Oblimin Criterion


Tl .58274 .32857 .41574 .42399

T8 .84855 .17963 .25808 .72114

T9 .85912 .16234 .20978 .74392

Tll .40259 .30678 .75676 .61679

T12 .16184 .06663 .88017 .79949

T14 .21744 .78306 .08341 .62293

T15 .01849 .78772 .14830 .65524

T16 .43034 .70050 .29993 .56755

(Sum =)

Trace 1.92663 1.77798 1.44683 5.15115

% of Variance 24.1 22.2 18.1 64.4

Note. Trace are computed differently than eigenvalues. To computethe trace for Factor I, each pattern coefficient is multiplied byits corresponding structure coefficient and then summed down therows. For Factor I, Trace = (.46919)(.58274)+(.85781)(.84855)+(.88892)(.85912)+(.15958)(.40259)+(-.10360)(.16184)+(.05184)(.21744)+(-.19885)(.01849)+(.24771)(.43034) = 5.15115. The

trace still sum to 5.15115, the sum of communality coefficients inthe Table 4 analysis.

31

Factor Rotation 31

Table 10

Factor Correlation Matrix after Rotation to the Oblimin Criterion

I

Factor 1

Factor 2

Factor 3

1.00000.24940.31242

1.00000.21387 1.00000

3?

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